Asked by yo
Given that (1, 0, 1) is a solution to a system of three linear equations, which of the following is true about the system?
The system can be either inconsistent or consistent.
The system can be either independent or dependent.
The system can only be independent and consistent.
The system can only be dependent and inconsistent.
All Answers 1
Answered by
GPT-5 mini
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Correct answer: The system can be either independent or dependent.
Reason: (1,0,1) being a solution means the system is consistent (has at least one solution), so it cannot be inconsistent. Whether the solution is unique (independent system) or nonunique (dependent system) depends on the equations' independence.
Examples:
- Independent (unique): x = 1, y = 0, z = 1 has the unique solution (1,0,1).
- Dependent (infinitely many): x + z = 2, y = 0, 2x + 2z = 4 — these are consistent but dependent, with (1,0,1) one of infinitely many solutions.
Reason: (1,0,1) being a solution means the system is consistent (has at least one solution), so it cannot be inconsistent. Whether the solution is unique (independent system) or nonunique (dependent system) depends on the equations' independence.
Examples:
- Independent (unique): x = 1, y = 0, z = 1 has the unique solution (1,0,1).
- Dependent (infinitely many): x + z = 2, y = 0, 2x + 2z = 4 — these are consistent but dependent, with (1,0,1) one of infinitely many solutions.
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